Flexible systems are difficult to control because they respond to human-operator commands with deflection and vibration that makes them difficult to move and accurately position. Input shaping is a control technique that reduces unwanted flexible responses by slightly modifying the human-operator command in real time. This modification is accomplished by convolving the unwanted flexible responses with a series of impulses called an input shaper. The input shaper is designed using estimates of the system's natural frequency and damping ratio. This process is demonstrated in FIG. 1 using a two-impulse sequence called a zero vibration (ZV) shaper. The upper left of FIG. 1 shows a pulse 10 in velocity and the position response 15 of a flexible system to the pulse command. When the pulse 10 is properly shaped by convolving it with the impulses 18, the staircase command 20 at the bottom of the figure is produced. Using the shaped command 20, the response 25 of the flexible system does not have residual vibration 28.
The penalty for the dramatic reduction in system vibration 28 can be a slight lengthening of the shaped command 20 by an amount equivalent to the duration of the input shaper (typically 0.5-1.5 vibration periods). This increase in command duration not only slightly increases the rise time of the shaped command 20, but can cause some additional system motion after the human operator has commanded the system to stop. This additional motion contributes to suppressing the vibration excited during the deceleration portion of the command 20. However, to accurately position a system, the human operator typically estimates the deceleration travel time and the resulting distance. This effect may make precise positioning of the system difficult, especially for inexperienced operators. Using longer duration shapers, such as those designed to be extremely robust to parameter variations or to eliminate multiple modes of vibration can exacerbate this effect.
FIG. 1 showed that a ZV-shaped pulse command 20 is longer than the unshaped reference command 10. In the figure, an ideal pulse 10 in velocity was shown. However, the velocity of real systems cannot typically be changed instantaneously. Therefore, velocity pulse commands are rate-limited and changed to a shape similar to trapezoids, where the positive and negative steps of the pulse are changed to ramps. The resulting trapezoidal profile can be categorized by its ramp time, which is the time it takes to accelerate to (and decelerate from) maximum velocity.